Self-sustained musical oscillators
Robert T. Schumacher · The Journal of the Acoustical Society of America · 1976
Most musically interesting oscillators consist of two systems, at least one of which is linear, coupled by a nonlinear force. Steady state oscillations of all such systems can be described by coupled nonlinear integral equations of the Hammerstein type, in which the kernels are determined by the impedances or admittances of the linear systems [R. T. Schumacher (to be published)]. Instruments may be classified as weakly nonlinear (woodwinds and brass played softly) or strongly nonlinear (bowed strings or beating reeds). Methods of solution differ strikingly in the two cases. In the former the reduction of the integral equations to a set of nonlinear algebraic equations to be solved numerically can be accomplished with the aid of computer programs for symbolic and algebraic manipulation. In the strongly nonlinear case, a generalization of Newton's method for a special model of a violin bow and a realistic string model has been successfully applied [R. T. Schumacher, op cit.] More generally useful and less expensive techniques depend on finding a suitable complete set of orthonormal functions for on-off processes (slip-stick or beating reed) and application of appropriate finite element methods of solution. Examples of solutions are given for chosen systems. [Work supported by NSF.]